2.1. Mechanic structure

As shown in Fig. 1, BPMs are fixed on rear of the solenoid. The 3D translation stage moves, driven by step motors, along the X, Y and Z axes with a precision of 1 μm in the closed-loop control. A BPM is small enough to be installed in a restricted space of the cryomodule. The vacuum aperture is 40 mm and the longitudinal length is 180 mm. The gap between the beam pipe and button along the longitudinal direction is 3.4 mm. We use four coupling electrodes to measure the horizontal and vertical position of the beam ^[11]. According to the equivalent capacity coupling model ^[6], Eq.(1) is used to calculate the normalized horizontal X and vertical Y positions with the mechanical center, where K_X and K_Y are the geometric coefficients of BPM and are the reciprocal of sensitivity (in mm); V_A–D are the voltages on the corresponding electrodes scaled with the appropriate calibration coefficients; and X_offset and Y_offset are the offset between mechanic and electronic center.

Fig. 1Full-screen View

Rotating workbench

The BPM measurement covers trajectory, first turn, equilibrium, turn-by-turn, beam-based alignment, and orbit stability ^[13,14]. To meet requirements of the superconductor cavity, button BPMs in cryomodule are expected to operate under ultra-high vacuum conditions, cryogenic environments, and a clean room class 100 certification. Hence a precise alignment is required for the BPMs, not only in the initial installation, but also a regular control of their actual position and the preservation of their nominal values over a long period. These are essential for an undisturbed operation of a machine or an experiment. The alignment procedure includes calibration, installation and monitoring.

2.2. Calibration

The calibration is to determinate the center of BPMs, reduce the coordinate error as much as possible, and transfer the data to external targets settled on the solenoid ^[15]. Fig. 2 shows the calibration bench, on which both mechanical and electrical methods are used. The mechanic method contains initial calibration of the individual solenoid, assembly, and calibration of the combined BPMs and solenoids. The initial calibration is performed by a portable coordinate measurement machine (CMM) with eight benchmark targets. The frame’s origin is the mechanic center of the combined BPM-solenoid. The Y-axis is purely vertical with opposite sign. The Z-axis is along the beam direction, and the X-axis is perpendicular to both the Y and Z axes. The signs are determined by the right-hand rule. Assembly the BPM in a clean room has an influence on the friction forces of bolts and nuts and the methodical steps of the assembly ^[11]. Therefore, to optimize the assembly and avoid deformation of the flanges from the excessive strength, according to simulation results, the torque is restricted to 5 N·m. The mechanic offsets for the assembly and alignment of BPMs are obtained on the designed workbench, as the error of fiducial target is not avoidable from the manufacturer ^[16].

Fig. 2Full-screen View

Calibration bench setup

The electrical calibration is to correlate the mechanic and electrical centers. It needs high performance of rotation symmetry, and the mechanic center should be aligned in advance to the center of rotating platform ^[17]. Therefore, the offset between mechanical and rotating center shall be tracked in the electrical calibration. Fig. 3 shows the calibration results of mapping functions using the difference over sum at the fundamental frequency of 162.5 MHz with a Libera Single Pass H, in a square region of ±4 mm lengths in 0.2-mm steps. The four output voltages picked up from the monitor are measured at 1681 wire positions in the horizontal and vertical directions like a lattice around the BPMs center. At each wire position, the pickup voltages are measured 50 times and the average value and the errors are stored together with the wire positions on a hard disk. In the central region of ±1 mm, the distortion was less than 10 μm, while it was 20 μm at the point of ±2 mm away from the center (the inside square in Fig. 3). This distortion is acceptable for our practical application.

Fig. 3Full-screen View

Mapping

2.3. Combined installation

The combined installation used a 3D coordinate transformation to uniform spatial frame. There are 3 models: Bursa, Molodensky, and Wu Ce. The purpose of coordinate transformation is to fulfill the start frame to the target frame with seven parameters. To restrict the displacement, rotation and scale errors within mm, 10⁻³" and 10⁻⁷, respectively ^[18], the 3D nonlinear model in Eq. (2) is used ^[19],

where, m is the scale factor, ω, φ and κ are rotation parameters, △x, △y and △z are displacement factors, and i is the point number (from 1 to n).

The BPMs are sealed in the cryomodule operated at 4 K, and the structure is complex, so multivariate total least squares (MTLS) is used to realize combined installment. To coordinate with a bigger rotation angle or under independent equal precision observation, MTLS is used for transformation with any scale factor ^[20], without the need of iterative calculation. In the MTLS solution, both the start and target data contain random errors, therefore, so the 6 DOF code (six degrees of freedom) is developed (Fig. 4) to limit calculation uncertainty and improve aligning accuracy. Two laser trackers in separate bilateral directions are used to improve the accuracy, avoid repeated stationing, and reduce the installation time.

Fig. 4Full-screen View

(color online) Solution of 6 DOF

2.4. Monitoring

A micro alignment telescope (MAT) was adopted to monitor vacuum and cooling down (77 K) displacements. Specific cross hairs was designed ^[21], and fiberglass (G11) was chosen as the material of cryo-crosshairs to verify the reproducibility of the horizontal and vertical position of the cold mass. As the aiming error of an optical instrument is proportional to the distance, we shortened the aiming distance and did multiple measurements to use the average ^[22]. To reduce the refractive error of glass, the incident angle was minimized ^[23]. It means that the aligning axis was parallel or coincident with the normal axis of the glass. Therefore, an electronic level was used for leveling MAT within 0.05 mm/m. To avoid diffractive error of light, cross hairs were arranged at different eccentricities (±0.2 mm). To facilitate the observation, cross hairs were rotated to different angles (±30/60 degrees). Prior to the monitoring, the laser tracker was used to adjust the two reference targets of the telescope to a straight line, and then, the MAT was aligned to the reference line using the “Far Rotation Near Translation” method ^[21]. Finally, the cross hairs of each BPM were aligned to the reference target line. In the monitoring process, as the deformation and displacement were beyond the telescope range (±1.2 mm), a dial indicator (Fig. 5) was used to reduce the uncertainty error to ±0.01 mm.

Fig. 5Full-screen View

Monitoring illustration used MAT

3.1. Results of simulating and monitoring

Usually, heat transfer includes the processes of thermal radiation, convection, and sometimes mass transfer. Being operated in cryo-vacuum environment, the BPMs themselves do not have convective heat transfer ^[24] except thermal radiation from the room environment, and thermal conduction through the cold mass supports and the feed-throughs. The finite element code of ANSYS was used to analyze the thermal stress and displacement, with Solid Works to construct model, meshing with four surfaces units. The cooling experiments were done with liquid nitrogen. The contact surface of support ball hinge was operated at 300 K. The surface heat load at 77 K (BPMs) was 0.9 W/m². The BPMs were made of stainless steel (AISI 316LN) with a weight of 5.2 kg. The loads included a self-gravity, a distributive load of temperature and the top suspending support. Temperature gradient in vertical direction is four times as much as in the horizontal direction, so the simulated horizontal and vertical cryo-displacements were 0.7 and 2.9 mm, respectively (Fig. 6a and 6b).

Fig. 6Full-screen View

(color online) Comparsion of the Cryo-deformation

Experiments were performed with five BPMs to monitor their cryo-deformations in horizontal and vertical directions. Fig.6(c) shows the results in a thermal cycling. As the target was located on the right (below) of solenoid, a plus sign means that it is close to center (rise up), and a minus sign means that it is off center (go down). After 24-h vacuum pumping, the BPMs were cooled down to 77 K, and in average they contracted 0.7 mm horizontally and 2.8 mm vertically. When the BPMs were warmed up to 290 K the vacuum is broken in 24 h, they expand in average by 0.5 mm horizontally and 1.4 mm vertically. The results indicate that the reproducibility is 0.2 mm horizontally and 1.4 mm vertically.

3.2. Results of Calibration and BBA

Fig. 7(a) shows the calibration results (five BPMs) using the “stretched wire” method, in which the BPM resolution was established by measuring the residual ^[25]. Electrical and mechanical reproducibility contribute to the residual offset ^[26]. Residual offset, i.e. the accuracy, is 0.02 mm (DX) and 0.01 mm(DY). A, Fig. 7(b) shows results of the five BPMs after combined installment, three displacement factors are <0.06 mm, and three rotated factors are <3 mrad.

Fig. 7Full-screen View

(color online) Alignment results

By correlating the magnetic center of a solenoid with a nearby BPM, we used the BBA procedure, a technique to derive the position of a quadrupole ^[27] or sextupole by detecting its orbit changes ^[28]. However, we applied the method to the solenoid by scanning the gradient of the field of solenoid one by one with a step of every 10 uA and recording all the readings of BPMs. Fig. 8 shows the offset of BPM electrical center with respect to the magnetic center of the solenoid. The blue lines are the nominal simulated value while the yellow lines are the actual measured value. As the accuracy was temperature dependent ^[29], and the cryo-deformation was nonlinearity from 300 K to 4 K, we could not simply sum all the contributions. Also, a solenoid is incompletely equivalent to a quadrupole or a sextupole. The energy and magnetic shift are nonlinear ^[30]. And there was the error in alignment of the solenoid and BPMs. Even though we eliminated the two BPMs, still there was a possibility that some BPMs worked abnormally at some specific measurements. Therefore, we obtained the offsets (0.1–0.2 mm) by comparing the measured orbit distortions with the simulated ones ^[27].

Fig. 8Full-screen View

(color online) BBA results